Efficient worm-like locomotion: slip and control of soft

 

Efficient worm-like locomotion: slip and control of soft-bodied peristaltic robots Kathryn A. Daltorio1, Alexander S. Boxerbaum2, Andrew D. Horchler1, Kendrick M. Shaw3, Hillel J. Chiel3, and Roger D. Quinn1 1

Department of Mechanical Engineering, Case Western Reserve University, 10900 Euclid Ave, Cleveland, Ohio 44106-7078, [email protected] 2 SRI International, 333 Ravenswood Avenue Menlo Park, CA 94025, [email protected] 3 Department of Biology, Department of Neurosciences, and the Department of Biomedical Engineering, Case Western Reserve University, 10900 Euclid Ave, Cleveland, Ohio 44106-7078 Abstract In this work, we present a dynamic simulation of an earthworm-like robot moving in a pipe with radially symmetric Coulomb friction contact. Under these conditions, peristaltic locomotion is efficient if slip is minimized. We characterize ways to reduce slip-related losses in a constant-radius pipe. Using these principles, we can design controllers that can navigate pipes even with a narrowing in radius. We propose a stable heteroclinic channel controller that takes advantage of contact force feedback on each segment. In an example narrowing pipe, this controller loses 40% less energy to slip compared to the best-fit sine wave controller. The peristaltic locomotion with feedback also has greater speed and more consistent forward progress. Keywords: Earthworm robots, peristaltic locomotion, ground contact modeling of soft robotics, biologically inspired control, stable heteroclinic channels 1. Introduction Legless animal-inspired locomotion can be ideal for navigating long narrow spaces. Worm-like robots can be compact enough to move through not only pipes (Yamashita, et al. 2011), but even through challenging medical environments such in intestinal tracts for non-invasive surgery (Glass, et al. 2008). In endoscopy, actuation at the tip can draw a scope further into the body than a traditional “flexible endoscope” could reach without incision (Yamamoto, et al. 2001). This is a slow process that would benefit from automation. Proposed robotic endoscopes typically alternate two types of segments, lengthextending and anchoring. The length-extending segments advance the tip with pneumatic bellows or linear motors. The anchoring segments hold the robot in place with radial legs, clamps, suckers, or passive spines (Dario, et al. 1999; Asari, et al. 2000; Cheng, et al. 2010; Bertetto & Ruggiu 2001, Wang & Yan 2007, Hoeg et al. 2000). More segments result in more traction (Slatkin, et al. 1995; Zuo, et al. 2005). In contrast, using one type of segment that performs both extension and anchoring may result in efficient, reduced-actuated soft-robots that can be built at smaller scales. Earthworms have one type of segment that does both axial extension and radial expansion. The body segments extend and expand in a process called persistalsis that moves the earthworm through its environment. While earthworm-like peristalsis is superficially similar to the rectilinear gait of snakes

  (Lissmann 1950) and snake-like robots (Tesch, et al. 2009; Hirose & Mori 2004; Transeth, et al. 2009), the difference is that earthworms have no rigid internal structures, but instead rely on a hydrostatic skeleton. In other words, earthworms are filled with incompressible fluid, so as a segment is constricted in diameter, it increases in length (Quillin 1998; Dorgan 2010). In earthworm peristaltic locomotion, a wave of radial contraction coupled with axial extension travels down the body, moving the body in the opposite direction of the wave’s travel. This coupling between segment diameter and length matters both for design and control. Radial expansion can affect the weight distribution when moving along a flat surface and will affect the pressure distribution when crawling in a constrained burrow or pipe. The ground reaction loads, in turn, affect the shape of the segments, which affects the loads and shapes of other segments. All these nonlinear interdependencies can make modeling and control complicated (Jones & Walker 2006). Nonetheless, compliance is compelling. Robots that are soft can adapt to and interact with the environment in ways that would be impossible for a rigid robot (Trivedi, et al. 2008; Trimmer, et al. 2006; Ingram 2006; Cowan & Walker 2008; Mazzolai, et al. 2012). Earthworm-inspired segmented robots have shown promising abilities to navigate straight pipes. One group of segmented robots can climb vertically in narrow spaces (Omori, et al. 2009), drill into dirt (Omori, et al. 2011) and measure pipe interiors (Yamashita, et al. 2011). Other groups have considered reduced actuation (Nakazato, et al. 2010) or actuation exterior to the pipe (Saga & Nakamura 2004). Soft body locomotion can be driven by coordinated oscillators. These oscillators are often based on limit cycles, systems of equations that stabilize into a regular periodic output. Changing the relative phase of an oscillator changes the locomotory gait (Ijspeert, et al. 2007). These controllers may be similar to the way animals control their bodies because groups of neurons connect to generate repeating cycles, often referred to as central pattern generators (CPGs). See (Ijspeert 2008) for a review of controllers inspired by this concept. Examples of robotic controllers include serpentine locomotion (Wu & Ma 2010; Conradt & Varshavskaya 2003) and lamprey-like locomotion (Crespi & Badertscher 2005). Adaptation of 4-segment undulation gaits is studied in (Schwebke & Behn, 2013). In addition to modulating the frequency and amplitude parameters over time, a low frequency step-function-like discrete controller can be added to the output (Degallier, et al. 2011). If there are many points along the body to be controlled, arrays can be built (Wilson & Cowan 1972; Wadden, et al. 1997). One of our previous papers was based on the WilsonCowan model to adjust the speed and spatial resolution of traveling waves (Boxerbaum, et al. 2011). We added feedback to limit radial expansion in (Boxerbaum, Daltorio, et al. 2012). A unique type of oscillator, based on mathematical structures called Stable Heteroclinic Channels (SHCs) may have some advantages. SHCs can produce regular periods of stable output (Guckenheimer & Holmes 1988; Afraimovich, et al. 2004; Laurent, et al. 2001). In other words, the system can “pause” near defined equilibrium points. This allows a designer to treat the equilibrium points as a sequence and yet the dynamic equations provide smooth continuous transitions. The transitions can be controlled by feedback terms. In this way, SHC pattern-generating networks can be varied to produce behavior ranging from the smooth robust outputs found in limit-cycle CPGs to the more sensory-gated state-like behavior found in finite state machines (Shaw, et al. 2012). Biological modeling applications of stable heteroclinic sequences and cycles have included olfactory processing in insects (M. Rabinovich, et al. 2001; Mikhail I Rabinovich, et al. 2008), the control of a neuromechanical limb (Spardy, et al. 2011), search behavior in the marine mollusk (P. Varona, et al. 2002), and the evolution of multispecies ecological systems (Afraimovich, et al. 2008). Engineering applications have included developing the control of a 2-D robotic manipulator (Verduzco & Alvarez 2000), control and decision-making algorithms for microspacecraft (McInnes & B. Brown 2010), identification of phoneme sequences for artificial speech recognition (Kiebel, et al. 2009), and a network of switching sequences to compute arbitrary logic operations (Neves & Timme 2009; Neves & Timme 2012). We have previously employed SHCs in a neuromechanical model of feeding behavior in the sea slug (Shaw, et al. 2010; Shaw, Cullins, et al. 2012) and in the PWM speed controller of a wheeled millipede-inspired robot (Webster, et al. 2013). In this paper, we will consider how this key feature of peristaltic locomotion, the coupling of significant changes in segment height and length, affects the design and control of peristaltic worm-like

  robots in a dynamic simulation. We generalize our analysis by considering fundamental energy terms. Evaluating existing peristaltic robots that use this coupling (Table 1) helps to determine that the key design constraint is slip. We have built a simulation of a simplified worm robot body to explore the effect of coupling on slip. The new simulation incorporates radially symmetric ground contact forces (such as in a rigid pipe or tunnel), as well as segment inertia, actuation force, and inter-segment deformation coupling. From this simulation, we identified two ways to reduce slip in a controller: wavelength control and contact force control. Finally, these control ideas have been implemented in a new SHC controller that takes advantage of the coupling to reduce slip, by using force feedback from each segment. Without force feedback, the output of the resulting SHC controller is similar to a clipped sine wave and effective in a straight constant-radius pipe. We then consider pipes with a decrease in radius. We refer to this constriction in the pipe as a narrowing. With force feedback, this controller allows our simulated robot to navigate a narrowing with only 60% of the frictional losses of a similar clipped sine wave controller. 2. Previous Work on Peristalsis In order to build our simulation, we will examine the fundamental energy terms inherent in wormlike peristaltic locomotion and briefly review some published robots. This review suggests that our simulation should model ground reaction forces and slip. We believe that our recent robot, Softworm, is unique among these robots, and raises possibilities for efficient motion worth exploring in simulation. In locomotion, efficiency is fundamentally limited by unrecoverable energy losses. In Alexander’s comprehensive comparison of locomotion types, hopping, walking, and running were shown to have no inherent frictional energy losses (Alexander 2003). However, he states that friction losses dominate total energy losses in low-speed earthworm peristalsis (Alexander 2003). Because he assumes that weight distribution among the segments is constant, the friction losses are equivalent to the weight of the body being dragged along the ground for the entire distance of travel (Alexander 2003). Under those assumptions, the the ratio of backward-sliding to forward-sliding friction coefficients limits the ratio of forward-moving to stationary anchoring segments. This limitation has been cited in the robot design process (Dorgan 2010). As a result, worm robots designs often only have more than half the segments in motion if there are anisotropic friction mechanisms. Examples of anisotropic feet, fins, or legs are found in (Schulke, et al. 2011; Menciassi, et al. 2004; Vaidyanathan, et al. 2000; Cheng, et al. 2010; Zimmermann, Zeidis & Behn, 2009). Another design uses retractable clamping feet (Zarrouk & Shoham 2012a), which results in heightlength coupling, although the robot was implemented with two different types of segment. If the friction is isotropic, the ratio of backward to forward friction is one, meaning that less than 50% of the segments can move forward at any instant, while the remaining segments are stationary anchors, as in (Seok, et al. 2010; Wang, et al. 2008). Table 1. Some compliant peristaltic robots that use a single type of segment to extend and anchor Non-anchoring Speed Slip Reference Segments Segments Efficiency (Vaidyanathan, et al. 2000) 3 spikes on foam, underwater 33 – 66%  0

𝑥!! = 𝑥!!!!!

!𝑇!! ! <   𝜇!"#"$% !𝑁!! !

!(2𝑙 ! )! − !𝑙!! !! = 2𝑟!𝑥̅!! !

!(2𝑙 ! )! − !𝑙!! !! = 2𝑟!𝑥̅!! ! 𝑇!!   =   𝜇!"#$#%& !𝑁!! !

4. Kinetic friction: sliding right

!(2𝑙 ! )! − !𝑙!! !! < 2𝑟!𝑥̅!! !

! !!!! 𝑥!!! = 𝑥!!!

𝑇!!   =  0 3. Kinetic friction: sliding left

Conditions

!(2𝑙 ! )! − !𝑙!! !! = 2𝑟!𝑥̅!! ! 𝑇!!   =

−  𝜇!"#$#%& !𝑁!! !

𝑁!! >  0 only used when last state free and 𝑥̅!! ≈ 𝑥̅ !!!!! 𝑁!! >  0 𝑥̅!! < 𝑥̅!!!! 𝑁!! >  0 𝑥̅!! > 𝑥̅!!!!

Note that friction constraint colors coordinate with Appendix Figure 12. Contact modeling is implemented numerically in different ways across software packages (Lu 2013). Penalty methods insert springs at the impact points to quickly approximate complex local deformations (Chatterjee 1997), which is an especially good approximation for a soft ground and rigid body (Azad & Featherstone 2010; Allotey & El Naggar 2008). For situations of sustained dry contact, Coulomb friction equations are best, but there are inherent discontinuities between static and kinetic friction. The discontinuities can be handled in various continuous ways (Alart & Curnier 1991; Bender & Schmitt 2006). For a review of numerical contact algorithms, see (Wriggers 1995). Finding a good way to insure fast and robust contact modeling is especially challenging for multiple contact points (Bender & Schmitt 2006). For this problem, we were able to satisfy Coulomb friction equations at each timestep with the heuristics described in the Appendix. The parameters used for these calculations and in the equations below are listed in Table 3 and compared with measured values for our robot Softworm. The number of segments is the same as the number of cable-actuated segments in Softworm. The maximum length is the same, but because our segment consists of only one rhombus instead of a mesh of rhombuses, the segment height, ℎ!! , of the simulated worm robot is less than the total diameter of Softworm (Figure 3). The simulation is capable of greater changes in segment length, since the actuation is not constrained by the dimensions of the cam mechanism. Thus, it travels faster when controlled at the same wave frequency.

  Table 3. Parameters Names and Values Used Set value, nominal range, or definition and sign convention Worm Body Measurements 𝑛, number of worm segments 𝑙 ! , link length 𝑙!! , segment length ℎ!! , segment height M, total mass 𝑚, nodal mass 𝐾, individual segment stiffness scale 𝑘, segment coupling stiffness 𝜇!"#$#%& , coefficient of friction 𝜇!"#"$% , coefficient of friction

Softworm 12 cable-tensioned segments 55 mm 54 mm to 79 mm 120 mm to 180 mm 4 kg 0.3 kg = 𝑀/(𝑛 + 1) body stiffness (315 N/m)/12 2.3 N/rad, 6 strands bending 0.1 against tile floor slightly greater than 𝜇!"#$#%&

Simulation 12 DOF 40 mm, unstressed segment length range: 25 mm (𝑙 !"#$%!$ ) to 79 mm range: 1.3 mm to 76 mm (pipe 2𝑟) 1.3 kg = 𝑚(𝑛 + 1) 0.1 kg 13 N/m 1 N/rad 0.1 0.12

Simulation Variables 𝑡, time 𝑥!! , node location along pipe 𝑥̅!! , potential contact point location 𝑠!! , contact point slip over timestep

trial duration: about 14 sec, max timestep 0.005 sec primary physical state variable for node 𝑖 at time 𝑡 ! mean of adjacent node locations: 𝑥̅!! = !𝑥!!! + 𝑥!! !⁄2 !

! !!∆! ! 𝑠!! =   𝑥̅!! − 𝑥̅!!!∆! = !!𝑥!! − 𝑥!!!∆! + 𝑥!!! − 𝑥!!!

𝜃!! , torsion spring angle between links 𝜃! = cos!! !!!!!! − cos!! ! !! !, torsion energy = ∑! 𝑘𝜃! ! /2 !!  ! !! ! !! ! ! COM, center of mass mean of all node locations, 𝑥! , at any time, COM =   ∑!!!! 𝑥!! /𝑛 unused, gravity forces neglected for radial symmetry 𝑔, gravitational constant 38 mm (nominal), 27 mm (minimum) in 316 mm-long narrowing 𝑟(𝑥), pipe radius ! range: 0 (inactive) to 1 (active) 𝑎! , SHC node activation 𝜂!! , SHC noise scaling parameter range: 10e−12 (nominal) to almost 10e–1 at high contact impacts 𝑓 ̅(𝑡), mean controller wave frequency about 0.95 Hz (similar to Softworm: 54 rpm = 0.9 Hz) range: –80 N to 50 N, positive increases segment length 𝐹!! , Force from vertical actuator ! mean non-zero: 70 N, maximum: ~1,000 N, positive compresses 𝑁! , Normal contact force from pipe mean abs non-zero: 6 N, positive pushes segments forward (right) 𝑇!! , Tangential friction contact force 𝑆(𝑙), segment spring force Nonlinearity bounds lengths, 𝑆(𝑙) = 𝐾  tan  (. 5𝜋(𝑙 − 𝑙 ! )/𝑙 ! ) scalar 200, lower values slow transitions 𝛼, SHC activity growth rate 𝜌, SHC connection matrix 𝑛-by-𝑛 circulant, nonsymmetric matrix relating node activations 𝛾, 𝜁 values in the connection matrix 𝛾 > 𝜁, in our case 𝛾 = 3  , 𝜁 = 0.025 sampled normal distribution with mean 0 and variance 1 𝒩!! (0, 1), random noise 𝐺, actuation force increase at contact 0 at posterior linearly increasing to 20N at anterior  𝐽(𝑙! ) Jacobian diagonal term Approximation of how actuator affects length 𝐽! ≈ 𝜕𝑙! ⁄𝜕𝐹! 10e-12 𝜀, minimum SHC noise scaling 𝛢! , 𝛢! , index of anchoring segments ideally evenly spaced, i.e, |𝛢! − 𝛢! | = 𝑛/2 ideal force range in straight pipe, 100N 𝑅, Normal force nominal range ! , Normal force set point desired force at contact for proportional control, 15 N 𝑁 𝑃! , 𝑃! proportional controller constants 𝑃! = 5e6 N/(m-sec), 𝑃! = 10 Hz Performance metrics Best Controller speed, rate of change of COM 160 mm/sec straight speed when COM in narrowing 110 mm/sec ! 9 Nm over trial 𝐿 , cumulative slip loss COT, cost of transport, energy per distance per mass steady state COT in straight pipe 0.1 J/kg-m or m/sec2 max COT over 1 sec in narrowing 10 J/kg-m or m/sec2

Sine Controller 100 mm/sec straight 90 mm/sec 15 Nm over trial 0.4 J/kg-m or m/sec2 > 30 J/kg-m or m/sec2

Softworm 66 mm/sec

  3.7. Relationship of simulation to Softworm robot To summarize, the robot Softworm has only one actuated degree of freedom, since all the segments are connected to a single motor and cam mechanism. The simulated worm robot has 12 actuated segments, each with one degree of freedom: length. In the robot Softworm, a section of the body increases in length as it decreases in diameter, because the body consists of a cylindrical mesh of hollow tube diamond shapes (Boxerbaum, et al. 2012). In our simulation, the segment length is coupled to height by a single rigid rhombus linkage. We model a line of 13 nodes, each with the same mass, with a segment defined between each pair of nodes. We assume that segments have stiffness that resists changes in length and a lesser stiffness that resists differences in adjacent segment heights. Given actuation forces, our simulation determines Coulomb friction pipe ground contact forces and node locations. Segment lengths and segment heights are derived from node locations. The robot Softworm moves on flat floors, with the sum of the ground reaction forces equal to the weight. In some configurations, the robot can tip slightly due to gravity when the front segments are extended forward. To focus on the effects of the height and width coupling, we have neglected this effect in our simulation. In future work, we hope to model segments with two degrees of freedom: length and tilt angle. 4. Control Objectives 4.1. Controllers evaluated based on frictional energy losses An ineffective worm controller will fail to make progress despite great exertion. To quantify the behavior of our controller, we will use a common metric, cost of transport. Cost of transport (Tucker 1970; Alexander 2003) is the energy expenditure per unit mass and per unit distance traveled. Note that the energy in the cost of transport (COT) can be normalized by weight (Tucker 1970) or by mass (Alexander 2003). Dividing by mass makes more sense for our simulated worm robots in pipes because we will be neglecting gravitational and buoyancy effects for radial symmetry. Lower COT indicates a more efficient motion. Instead of energy expended, we will look at energy lost. Energy loss agrees well with actuator work (Figure 10), and can be compared for any design regardless of actuator efficiency. For slow crawling animals, the largest energy losses are from friction with the ground (Alexander 2003). Consider our simulated worm robot, a series of n segments, each with index 𝑖 and mid-point location ! 𝑥! at time t. The segments can contact the interior pipe surface only at the midpoint locations. Neglecting gravity, we can assume that the pressure on the interior wall is radially symmetric for each segment. In two dimensions, a planar robot in a pipe would have equal and opposite normal forces, 𝑁! , on the upper and lower pipe wall. It follows that the tangential friction forces, 𝑇!! , are equal if the friction properties are radially symmetric. The simulation determines a time series of friction forces and the path of the midpoint locations (potential contact points) over time. If the contact points slip while applying these friction forces, energy is lost. The cumulative slip energy loss, 𝐿, is the work done against friction. The output of the simulation can be used to approximate this path integral: ! !

 

𝐿 =− !!! !"#$

!

2𝑇!!

d𝑥!!

! !!!! ! 𝑇!! + 𝑇!!!!! 𝑥!!!!! − 𝑥!! + 𝑥!!! − 𝑥!!! /2                      (4)

≈ !!! !!!!

For segments not in contact with the ground, 𝑇 = 0 (see Table 2, state 0). However, if none of the segments touches the ground, the simulated worm robot will not accelerate. To use ground traction, our controllers will expand some segments until they are as large as the pipe radius. 4.2. Controller should be able to maintain two non-contiguous anchor points The next design question is: how many segments should contact the pipe at any given time? There is a trade-off: more segments in contact with the ground may provide redundancy and reduce slip. However, keeping more segments stationary lowers the maximum speed, so having fewer segments in contact with the ground may allow the robot to go faster. An earthworm will typically use contiguous groups of 10 or

  more segments of the total 150 segments (Gray & Lissmann 1938). This may help them to distribute their weight or radial pressure loads and ensure traction. However, worm robots tend to have fewer total segments (Table I). For example, one of our previous robots had only three segments and slipped backwards between the anchoring of the front and rearmost segments (Mangan, et al. 2002). Adding more segments, allows the front most and rearmost segments to be in contact at the same time as the robot moves forward. With four segments, the first and fourth segments can be in contact with the ground and not slipping (anchored). The third segment elongates while the second segment shortens in length. As the second segment shortens in length, it increases in height until it contacts the ground. Another factor is that if a robot is moving on flat ground, it may tilt under gravity if there is only one contact point in the center. It is for this reason that Softworm has two waves per body-length. These examples demonstrate that either in a pipe or over open ground, a controller should be able to handle multiple regions of contact along the body while using inter-contact segments to progress. For this paper, we will design our controllers to have at least one contacting segment per wave, and two waves per body. Two nonadjacent contact segments is the minimum number of anchoring segments. Note that anchoring segments will refer to the intent of the controller to fix certain segments and contacting segments will refer to the calculated contact state of the segment. In other words, the anchoring segments are the segments that the controller is attempting to bring into a static friction state from table (2). Anchoring and contacting segments are coincident for a well-controlled robot on flat ground or constant diameter pipe, but extra segments can make contact with the interior of a narrowing pipe. 4.3. Anterior segments set up posterior contact locations Consider the configuration of the simulated worm robot in Figure 4. Assuming zero slip, the contact point of the segment posterior to the current contacting segment will be determined by the length of the segments at the contact height. In fact, all the contact points (black dots in Figure 4) are predetermined between the anchor points because of the coupling of the segment retraction and radial expansion. This means for a wave traveling along the body of the worm, the stride length is fixed posterior to an anchoring segment. If the two anchoring segments bracketing a group of non-contacting segments (i.e., a a single inter-contact length (Figure 4)) have indexes 𝐴!  and 𝐴! , the “stride length” is the in-pipe wavelength (the sum of current segment lengths 𝑙!! ) minus the sum of the future contact lengths 𝑙!!"#$%!$ of the segments in the wave. !!

!!

𝑙!!

Stride  length = !!!! !!

𝑙!!"#$%!$                                                                                      (5)

− !!!! !!

The in-pipe wavelength at the anterior of the robot must be large enough that the stride length is positive. This is trivial in the case of a constant-radius pipe since the contact length will be less then noncontacting lengths. However, in a narrowing pipe, it is possible for the future contact lengths to be less than current non-contacting lengths, so care must be taken to extend non-contacting segments sufficiently in order to progress. After the front segment contacts the pipe and becomes an anchoring segment, the in-pipe wavelength must be such that the anchor points do not slip. If the in-pipe wavelength varies for given anchoring segments, slippage must occur. There are two strategies to reduce slip: first, control of the inter-contact segments (wavelength control) and second, control of the contacting segments (traction control).

 

Figure 4. The body of the simulated worm robot (green) between the walls of a pipe (gray) at one time-step in the simulation. The extension of the rhombus segments is determined by the combined effects of contact forces from the interior pipe wall and actuator forces. Actuator forces are shown in the graph at the top aligned above the body segment at which they are applied. As expected, the body segments are shorter where the actuator forces are more negative and longer where they are positive. The contact forces are represented as bars above and below the pipe wall. Only segments in contact with the wall have non-zero contact forces. The normal component (black) of the contact force is much greater than the tangential friction components. The tangential components of the force are plotted above the normal force: blue is negative, i.e. the force on the body is to the left, and pink is positive, i.e. the force on the body is to the right. To understand peristaltic motion, it may be helpful to consider the length of the body between contact points (segments 4 to 10). To move forward in the anterior direction, the segments posterior (segments 3 and 9) to the contacting segments (segments 4 and 10), will come in contact next by retracting in length and increasing in height. The retraction of segment 9 is defined by a decrease in segment length, but the total inter-contact length cannot change without slip. To balance this retraction such that the contact points do not slip, another segment (segment 5) between the contact points must increase in length (extend), such that the inter-contact length is held constant. Similarly, segments (segments 6 and 8) two away from each contact point are paired with matched changes in length. The segment in the middle (segment 7) should be at its maximum length. Eventually, if there is no slip, each segment will contact the pipe at points (black dots) spaced by the contact length from the adjacent contact point. Meanwhile, the current contacting segments will elongate, losing contact. The crimson dashed line is the future path of the upper point of the posterior contacting segment (segment 4) assuming no slip. The posterior contacting segment (segment 4) will again touch the pipe wall at six contact lengths from the anterior contact point (segment 10) because there are six segments per wave here. It would be possible to have as few as two advancing segments between contacting segments (one to extend and one to retract). However, if there were only one extending segment between two contacting segments, the inter-contact length would have to change as the segment extended and the contacting segments would necessarily slip. If this single segment did not extend or contact, it would be unable to change diameter in order to contact the pipe before the release of the original two contacting segments.

  4.4. Wavelength control to reduce slip loss First, the control of the inter-contact segments should reduce the necessary friction force by maintaining the proper net inter-contact length. We will call this wavelength control. Consider the “free” condition of the robot in which actuator forces, F, but no pipe ground reaction forces act on the segments. Compared to the free condition, the “in-pipe” condition will decrease the radius and increase the length of any contacting segments. The pipe radius will determine contacting segment geometry. For the other segments, the in-pipe length is not constrained and may be different than the free length. Compressing or extending a series of non-contacting segments to be shorter or longer than their net free length requires force in the axial direction. In the case of free segments bracketed by anchoring segments, as in the intercontact length in Figure 4, these forces can come from friction with the pipe wall. In contrast, if the free inter-contact length is close to the in-pipe inter-contact length, minimal friction forces will be required to maintain the existing contact points. To make progress, some segments extend while others retract such that the sum of the lengths is constant and equal to the inter-contact length and the net balance of forces transmitted is minimized. Even before the contact point shifts from one segment to the next, each segment midpoint, 𝑥! , increases while the inter-contact length, remains constant. Eventually, the retracting segments’ radii increase sufficiently to contact the ground and become the new contact points. Mathematically, under certain simplified and ideal conditions, the accelerations and decelerations of individual segments can cancel and require no net friction force (Boxerbaum, Shaw, et al. 2012) (Keller & Falkovitz 1983). In our present segmented worm simulation, we find that if waves have inter-contact lengths close to the sum of the free lengths of the segments between anchoring points, progress is smoother. For this reason, we are careful in choosing initial conditions to reduce settling time. We gradually increase the actuator force amplitude so that none of the segments “stick” in a disadvantageous initial configuration. This ramp-up time is visible in Figure 11, but in Figures 8, 9, and 12, the time axis begins after several cycles. 4.5. Traction control to reduce slip loss The second way to reduce slip is to firmly anchor the contact points with sufficient normal force. We will call this traction control. This is the strategy often adopted in endoscope design: dedicated clamping actuators to generate large normal forces (Dario, et al. 1999; Asari, et al. 2000; Cheng, et al. 2010; Bertetto & Ruggiu 2001). Because of the axial-radial coupling, we can use the same segments to both progress forward and anchor. By anchoring firmly, any differences between the in-pipe inter-contact length and free inter-contact length can be absorbed by the compliance of the body and supported by the friction forces. Since these are relatively soft bodies, this friction, and thus the required normal force, need not be great. However, there might be payload (e.g., the dragging end piece in Softworm or a tether), sudden environmental shifts, or viscous drag that would make it valuable to maintain positive normal force so that friction forces are not limited. The prototype endoscopes with two different types of actuators tend to have clamping and unclamping states (Hoeg, et al. 2000). In analyzing rigid screw-based robots, it is helpful to categorize segments as either in extension, clamping, or partial clamping states in a compliant tube (Zarrouk, et al. 2010). Therefore, we want to design a controller for discrete worm segments that causes segments to pass through extension and retraction states (which balance each other between contact points) and anchoring states that control for a desired contact normal force. Note, however, that excessive contact force can increase slip loss when slip does occur. For example, if pipe radius is not constant, then the segments that should be freely extending encounter the pipe wall. These unexpected contacts can cause slip. Anisotropic friction, like that produced by worm setae (Murakami, et al. 2006), or their mechanical equivalents in robots (Menciassi, et al. 2004; Vaidyanathan, et al. 2000; Schulke, et al. 2011) help the body progress forward instead of backward when the no-slip condition cannot be maintained.

  5. A Stable Heteroclinic Channel Controller for Peristalsis The controller determines the forces, 𝐹!! , generated by the actuators on the individual body segments. We will compare three different controllers of increasing complexity, Figure 5: (A) the force is a clipped sine wave, (B) the force is controlled by a pair of control modes (anchoring and progress by extension and retraction) and (C) those same control modes with timing based on ground contact feedback. In (B) and (C), each segment actuator force is controlled one of the two modes that switch in a traveling wave down the body. The traveling wave required for (B) and (C) is generated with a stable heteroclinic channel (SHC).

Figure 5. Three controllers were evaluated. Controller (A) is an open-loop clipped sine wave. Controllers (B) and (C) use an actuator controller with mutually exclusive modes for anchoring or progression for each segment. The sequence and phase is determined by stable heteroclinic channel (SHC) dynamics. Feedback is used to ensure that the anchoring states make good contact. In (C) the physics affect the pacing of the SHC. The contact forces modulate the SHC output, determining which SHC node is active and thus which segment should anchor. 5.1. Controller A: Sine wave controller The sine wave controller is most like the kinematic output of Softworm’s cam mechanism. In Softworm, as the cam rotates, six pairs of cables attached to the segments of the worm are pulled in sequence. At any given time, the series of segments along the length of the robot are extended based on six evenly spaced rotations of the cam. An individual segment experiences a continuous cycle of all possible rotation actuations within the period of cam rotation. In controller (A), as in Softworm, each discrete segment gets the same smooth waveform, shifted by a fixed time, assuming that local strain is a function only of local actuator stress, as prescribed by (Boxerbaum, Shaw, et al. 2012). Initially, we used a zero mean sine wave that induced good simulated motion in a straight pipe. However, the output of our best controller (C) looked more like a clipped non-zero mean sine wave. This was because we limited the maximum actuator values so as not to overextend segments. After designing that controller, we did a regression fit of the output of the (C) controller to get a wave with mean at 10% of the sine wave amplitude and a maximum value clipped at 70% of the amplitude: Controller  A:        𝐹!! = min 50, 70 sin 50

2𝜋 𝑖 + 0.95 2𝜋 𝑡 + 7.5                                                                  (6) 𝑛

Here, 𝐹!! (in Newtons) represents the force of the actuator for segment i at time t, n is the total number of segments, and 𝑙 ! is the unstressed segment length. The negative values, corresponding to segment retraction and anchoring, were unclipped. See Figure 11 for an example wave. This regression-fitted

  clipped sine wave worked better than the initial zero-mean sine wave because it had larger inter-contactlength (and thus greater stride length and higher speed) with similar ground reaction forces. 5.2. Controller B: Anchoring and Extension/Retraction Cycle based on blind SHC As explained in Section 4, segments should extend, then retract, and then anchor to the wall. Thus there are two independent control objectives (traction control for contacting segments and wavelength control for moving segments) that have been implemented as two control modes: the anchor mode and the progress mode with paired extension and retraction. Each segment is controlled by one of the two modes. Unlike finite state machines, smooth mode transitions are desired. Limit cycles commonly used in CPGs produce smooth outputs akin to sine waves in which the amplitude is at the maximum value for only an instant. Since we want the contacting segments to stay firmly anchored as the other segments progress, we could threshold this output as in the clipped sine wave controller (A). Alternatively, we could retune the 25 scaling parameters of our previous 150 segment model (Boxerbaum, et al. 2011) that modulated the amplitude in response to feedback (Boxerbaum, Daltorio, et al. 2012). However, we want variable dwell times at phases in the oscillation cycle in response to environmental feedback, so it makes sense to use a different mathematical oscillator: stable heteroclinic channels. Stable heteroclinic channels (SHCs) are a framework for continuous dynamic oscillation that can produce regular periods of stable output. The underlying dynamical structure is based on the principle of “winnerless competition” (Laurent, et al. 2001; Afraimovich, et al. 2004). Conceptually, an SHC consists of a sequence of saddle points, which are equilibrium points that attract trajectories in certain directions (referred to as a stable manifold), and repel trajectories in others (an unstable manifold). By setting up the unstable manifold of one saddle to lead directly into the stable manifold of another saddle, sequences of transitions between the saddles can be generated. Paths in phase space connecting two different equilibrium points are called heteroclinic connections which can be connected into networks, i.e. SHCs. We will call the equilibrium points on this path SHC nodes. (Guckenheimer & Holmes 1988) first found stable (attracting) cycles of heteroclinic channels in turbulent flow equations. Here we use Lotka-Volterra equations (Gause 1932; Gilpin 1975; May & Leonard 1975; V. Afraimovich, et al. 2008), but these systems of saddle equilibria can be produced by a variety of models ranging from a chain of piecewisecontinuous linear saddles (Shaw, et al. 2012) to pulse-coupled oscillator networks (Neves & Timme 2009) to networks of Hodgkin-Huxley neurons (Nowotny & Rabinovich 2007). The GuckenheimerHolmes system (Guckenheimer & Holmes 1988), which has a mathematical structure that closely resembles the Lotka-Volterra system, is also commonly used (Ashwin & Karabacak 2011; McInnes & B. Brown 2010; Li, et al. 2012). We use a modified version of the Lotka-Volterra equations for our controller in the form of the following system of stochastic differential equations with 12-dimensional state variable 𝑎: !

d𝑎!!

=   𝑎!!

𝜌!" 𝑎!! d𝑡 + 𝜂!! d𝑊!!                                                                                                (7)

𝛼− !!!

Here 𝑎!! is the component of the state variable in the ith dimension as a function of time, 𝑡. There is an SHC node, i.e. a saddle point in state space, that corresponds to each dimension. We define 𝑎!! as the activation of the ith SHC node. 𝑎!! ranges from 0 to 1. SHC nodes with activity near 1 are designated active and SHC nodes with activity near 0 are inactive. 𝛼 = 200 is the instantaneous activity growth rate, 𝜂!! is a parameter controlling noise levels over time, 𝑊!! is an n-dimensional Wiener process, and 𝜌 is a connection matrix:

  1 0 𝛾 𝛾 𝛾 𝜁 𝜌 = 𝛼   0 0 𝛾 𝛾 𝛾 𝛾

𝛾 1 0 𝛾 𝛾 𝛾 𝜁 0 0 𝛾 𝛾 𝛾

𝛾 𝛾 1 0 𝛾 𝛾 𝛾 𝜁 0 0 𝛾 𝛾

𝛾 𝛾 𝛾 1 0 𝛾 𝛾 𝛾 𝜁 0 0 𝛾

𝛾 𝛾 𝛾 𝛾 1 0 𝛾 𝛾 𝛾 𝜁 0 0

0 𝛾 𝛾 𝛾 𝛾 1 0 𝛾 𝛾 𝛾 𝜁 0

0 0 𝛾 𝛾 𝛾 𝛾 1 0 𝛾 𝛾 𝛾 𝜁

𝜁 0 0 𝛾 𝛾 𝛾 𝛾 1 0 𝛾 𝛾 𝛾

𝛾 𝜁 0 0 𝛾 𝛾 𝛾 𝛾 1 0 𝛾 𝛾

𝛾 𝛾 𝜁 0 0 𝛾 𝛾 𝛾 𝛾 1 0 𝛾

𝛾 𝛾 𝛾 𝜁 0 0 𝛾 𝛾 𝛾 𝛾 1 0

!

0 𝛾 𝛾 𝛾 𝜁 0                                                                (8) 0 𝛾 𝛾 𝛾 𝛾 1

where 𝜁 =  0.025 and 𝛾 = 3. Note that the matrix is circulant, i.e. the diagonals wrap around, and thus each SHC node has identical connections to other nodes based on relative location. We integrate (7) numerically using an Euler-Maruyama scheme, resulting in the following system of difference equations: !

∆𝑎!!!∆!

=   𝑎!!

𝜌!" 𝑎!! ∆𝑡 + 𝜂!! ∆𝑡  𝒩!! 0, 1                                                          (9)

𝛼− !!!

The SHC state variable 𝑎! is augmented by ∆𝑎 at timesteps, ∆𝑡 ≈ 0.001 sec. The timestep size is approximate because smaller timesteps are used when multiple contacts must be resolved (as described in Section 3 and in the Appendix). Wiener process noise is time-invariant so different size time-steps can be used. Also the state variable 𝑎! is prevented from exceeding boundary conditions as the noisy forcing might perturb the state, 𝑎! , outside of the valid regime (0 ≤ 𝑎! ≤ 1). Here 𝒩!! 0, 1 is a normally distributed random number with zero mean and unit variance. Scaling the normal variates by the square root of the timestep ∆𝑡 makes the stochastic term act as a time-scale-invariant Wiener process noise. As a consequence, it is a good model for continuous Levy walks such as Brownian motion or zero-mean memory-less white noise. The noise term is explored further in the next section. The parameter values of the connection matrix, 𝜌, were chosen using a Matlab toolbox developed for the design, analysis, and simulation of SHC networks (Horchler 2013). Equation (7) defines an SHC manifold with as many dimensions as segments. The activity of each SHC node grows exponentially in a stochastic manner. When properly tuned, the activation value is quite small for most SHC nodes and close to 1 for the other SHC nodes. We will call SHC nodes with activation more than 0.9 “active.” Active SHC nodes are colored red in Figure 6. The stochastic growth is limited by inhibition from other nodes. Each SHC node inhibits the activation of most of its nearby neighbors (wrapping around the ends of the simulated worm as shown by the dashed lines in Figure 6). However, the SHC node immediately posterior to a given SHC node is not inhibited, and thus the activity of that SHC node slowly grows until it is large enough to inhibit the SHC node that was originally most active. This is a class of Lotka-Volterra population competition dynamics equations in which each segment population (SHC node) activity inhibits (“preys on”) the activity of other SHC nodes, except for the SHC node immediately posterior in the cycle. Eventually, the activity in this subsequent SHC node grows because the current active node does not inhibit it. It becomes the active SHC node and further inhibits the previous active SHC node. This creates the traveling wave that will drive the sequence of actuator extensions and retractions in order for our simulated worm robot to contact and follow the walls of the pipe. There are two waves per body because the inhibition only strongly affects four neighboring nodes ahead and behind it, so there are uninhibited nodes six segments away (half the body length away).

 

Figure 6. Each of the 12 body segments is associated with an SHC node. The activity of an SHC node inhibits its nearby neighbors, except the immediately posterior node (see the connection matrix (8)). This asymmetry causes the active SHC node to shift rearward as the body moves forward. In controllers (B) and (C), the segments in the inter-contact length are controlled by the extension/retraction mode because they are associated with the inactive nodes. The contacting segments are controlled by the anchoring mode because they are associated with the active SHC nodes. The normal forces (gray bars) scale the SHC node noise only in controller (C). The magnitude of the noise shortens or prolongs the period of the active node, (see Figure 7), thus keeping it aligned with each new contacting segment (see Figure 8). Segments corresponding to active SHC nodes (𝑎!! >.9) are controlled by the anchoring control mode to firmly expand against the pipe. When a segment’s SHC node is inactive, a weighted sum of the neighboring node activities is integrated to determine the control force (a progress extension/retraction mode of the actuator force controller). 5.2.1.  Anchoring  where  an  SHC  node  is  active   The anchoring mode generates and maintains ground contact, i.e. traction control from Section 4.5. On onset, when SHC node activity crosses the threshold 0.9, the actuator force at that segment, 𝐹! , is decremented by an initial amount. Then, if any slip is detected we further decrease the actuator force proportional to the magnitude of the slip. In the absence of slip, generally shortly after contact, the actuator force was controlled to maintain a chosen actuator force. Upon exiting the anchoring mode, the maximum of the current force or the onset force was augmented by the same amount. More precisely if there is some slip, 𝑠!! of the ith segment at time t: !

! !!∆! 𝑠!! =   ! 𝑥!! − 𝑥!!!∆! + 𝑥!!! − 𝑥!!!                                                                                     10 !!  !"#$% 𝐹!!  !"#$%& = 𝐹!!!∆! − 𝐺                                                                                                             11 ! 𝐹!!  !"#$%& = 𝐹!!!△! − 𝑃! s!!!∆! d𝑡 + 𝑃! s!!!∆! < ℰ!"# 𝑁!!!∆! − 𝑁 ∆𝑡                          (12) !!  !"" !!  !"#$% 𝐹!!  !"#$%& = max 𝐹!!!∆! − 𝐺, 𝐹!!  !"#!!"                                                                          (13)

where 𝐺 is a small initial offset, which varies linearly from 0 at the posterior to 20 Newtons at the anterior. Anchoring more at the front of the simulated worm robot is consistent with worm measurements (Quillin 1998) and seems to reduce backwards slip. 𝑃! and 𝑃! are constants, 𝑃! = 5e6 N/(m-sec) and 𝑃! =   10 N/(N-sec), respectively. We use Iverson bracket notation in (12) (i.e., s!!!∆! < ℰ!"# = 1 iff

  s!!!∆! < ℰ!"# is true, otherwise s!!!∆! < ℰ!"# = 0) and thus ℰ!"# = 0.001 mm is a tolerance that ensures the reaction to the normal force is not active when the contact point is slipping. 𝑁 is the normal force set point, 15 N. While not shown above, the actuator force, 𝐹! , is limited to a maximum of 50 N and the change in control force is also limited to a maximum of 10 N and a minimum of –5 N. 5.2.2.  Progression  by  paired  extension  and  retraction  where  SHC  nodes  are  inactive   We paired the extension and retraction within a single wavelength so that the segments of similar lengths (e.g., the segments before and after the contact point) would have matched length rates-of-change to maintain the inter-contact length. We refer to this as wavelength control in Section 4.4. In general, there are six segments in a wavelength: one contacting and two extending paired to two retracting. That leaves one segment unpaired in the middle. The middle segment should be at the maximum extension suitable for the segment, resulting in a larger inter-contact length than an unclipped sine wave. Having a larger vs. smaller inter-contact length is better for getting through pipe narrowings. In a narrowing, the contact length is greater. If the inter-contact length is too small, the stride length can go to zero, stalling progress (see equation (5)). Also, this middle segment acts as a buffer if the forward anchoring segment becomes inactive at a different time than the posterior anchoring segment. The SHC inhibition cycle specified by connection matrix (8) discourages less than five or more than seven segments between anchoring segments, but if even outside those bounds, this weighted integration would be robust. Thus, the actuator force 𝐹!! is determined by (11) when the associated SHC node first becomes active, by (12) during the active period when (𝑎!! > 0.9), and by (13) when the SHC node first becomes inactive. The rest of the time, the actuator forces are integrated numerically as ! ! 𝐹!!!∆! =   𝐹!! + 𝑅𝑓 0.6𝑎!⊕! − 0.6𝑎!⊖!

! 𝐽 𝑙!⊖! ! 𝐽 𝑙!⊕!

! ! + 0.4𝑎!⊕! − 0.4𝑎!⊖!

! 𝐽 𝑙!⊖! ! 𝐽 𝑙!⊕!

∆𝑡                  (14)

where 𝑅 is the nominal force range (100 N). 𝑓 is the controller wave frequency, with value of about 1 Hz, and it is recalculated at each timestep to reflect changes in average period. Specifically, 𝑓 is the inverse of the measured mean time periods between the onset of activity, 𝑎 > 0.9, averaged over all the segments. ⊕ is a circular shift operator. Thus, 𝑖 ⊕ 1 refers to the node 1 segment anterior to the 𝑖th node. It is a circular shift because we have connected the first and last SHC node to form a cycle. Similarly, ⊖ refers to a posterior circular shift in which the posterior node of the last segment is considered to be the first node. The terms for the immediate anterior and immediate posterior nodes are both multiplied by 0.6 in order to match extension and retraction. The second neighbors are multiplied by 1 – 0.6 = 0.4 in order that, over two time periods, the forces will increase by the full nominal force 𝑅, and then, over two time periods, decrease again before contact. 𝐽 𝑙! is an approximation of a diagonal term of the Jacobian of the segment length as affected by the actuator force, which we approximate via 𝐽 𝑙! = 𝐾 cos

𝑙! − 𝜋 2

𝑙!

!

≈

𝜕𝑙!                                                                                                          (15) 𝜕𝐹!

Because the segments are paired with similarly-sized segments, the Jacobians are nearly equal and ! ! ! ! 𝐽 𝑙!⊖! /𝐽 𝑙!⊕! ≈ 1 and 𝐽 𝑙!⊖! /𝐽 𝑙!⊕! ≈ 1. The duration of the active phase of each SHC node varies slightly due to the stochastic nature of the SHCs, but the system appears to be robust to this variability.

 

Figure 7. The SHC activation sequence with the noise magnitude, 𝜂 = 𝜀 =  10e– 12 (see (16)) results in SHC nodes being active (𝑎 > 0.9) for a mean period of 0.176 seconds. Changing the noise magnitude 𝜂 alters this mean period (Stone & Holmes 1990). A temporary increase in the noise magnitude 𝜂 prior to the activation of the subsequent SHC node can precipitate an early transition from the prior active SHC node. In this figure, for illustrative purposes, the noise magnitude 𝜂 is equal to ε = 10e– 12 at all times except during the brief interval indicated by the green bar when 𝜂 is set to 1. Note that the vertical lines are evenly spaced at the average activation duration. The mean phase of the overall system is unaffected by an early transition of one of the SHC nodes. In Controller (C), contact forces scale the noise magnitude 𝜂 so that early contact with the pipe by a body segment triggers a transition. 5.3. Controller C: Adding timing feedback. While (B) and (A) work well if designed for a particular pipe diameter, both are much less efficient in pipe narrowings. In such situations, segments touch the ground early, and thus cannot extend or retract properly, changing the free inter-segment length. This requires either slippage or more traction on the anchoring segments. However, a narrowing in a pipe should be an opportunity to get more horizontal traction per contact pressure because of the change in ground angle. This can be accomplished by using the SHC framework to allow segments to compete to be anchors. Controller (C) is different from (B) in that we provide feedback to the SHC used for timing. Controller (B) used a fixed noise magnitude, 𝜂 = 𝜀. In controller (C), the noise magnitude in each direction is scaled based on contact normal forces: 𝜂!! = 𝜀 + 10!! 𝑁!!!∆!

!.!

                                                                                                                             (16)

where 𝜀 = 10e– 12. The effect of the Gaussian noise on the mean first passage time for linear (and linearized) saddle systems has been characterized by (Stone & Holmes 1990). Figure 7 shows how adding more noise terminates the activity of a node more quickly. This gives an SHC node a competitive advantage in the competition to become active and hence anchor its associated segment. This makes sense because the ability to generate large contact normal forces is a mechanical advantage for an anchor point. The result is that the contact point is in phase with the active SHC node. There are two advantages to having the feedback in the stochastic term in (9) to modulate the period of the SHC. First, it suggests that this method will handle noisy sensor data well. Second, by isolating the control parameters in the stochastic term, we can use tools developed for characterizing similar stochastic differential equations.

  6. Performance The initial conditions in the simulation are such that the simulated worm robot is at rest and not touching the walls. As expected, the simulated worm robot does not start to progress until traction is gained. An ill-designed controller could result in no forward progress, backward motion, or constantvelocity floating without touching the walls. An effective controller will lead to the simulated worm robot settling into forward motion with small friction forces. We tested the three controllers discussed in Section 4 in a simulated pipe with a constant 0.38 m radius at first (𝑥 < 0.7  m), then a smooth narrowing and widening in radius, followed by a long constant 0.38 m radius pipe (after 𝑥 > 1.0  m). All three controllers were able to navigate this path. First, we will consider behavior in the straight portion of the pipe. The values of the key parameters after settling into motion in the long straight (constant-radius) pipe are plotted for the (C) controller in Figure 8. In the straight pipe, all three controllers develop similar normal forces and segment deformations. The (B) controller has longer and less regular activation times because it lacks feedback to terminate the SHC node activity when the next segment makes contact. The (A) control was fit to the (C) output forces in the 6th segment and phase-offset for each segment. Thus (A) and (C) have the same actuator force ranges. Second, we will consider the behavior in the pipe narrowing. In addition to the desired behavior in straight pipes, each of these controllers allows the simulated worm robot to pass through a 30% decrease in the pipe radius. The passage with the (C) controller is diagrammed in Figure 9. The simulation output is visualized in greater detail in Figure 12. Next we consider energy efficiency. As can been seen from Figure 10, most of the energy in the simulated worm robot is stored in the nonlinear springs, with a smaller amount stored in the torsional springs connecting segments and an even smaller amount is kinetic energy. From Figure 10 (middle panel), it is clear that friction energy loss correlates well with system energy and actuator work. The frictional cost of transport with respect to the total mass, 𝑀 = (𝑛 + 1)𝑚 is related to the frictional energy loss, 𝐿! defined in (4) 1 d𝐿!                                                                                                                                                                        (17) 𝑀 𝑑𝑥 which can be numerically differentiated (blue, Figure 10c) with timestep = 1 sec since wave frequency is 1 Hz. A possible worst-case estimate of a cost of transport would be the friction coefficient times the mean-over-time sum of all the normal forces divided by the motion of the center of mass (COM) COT =

!!

Worst  Case  Friction  COT  Estimate   =

2𝜇!"#$#%& 𝑀  

!!

!

𝑁!! ∆𝑡 ! /𝜏                                      (18)

  !  !! ! !! !!!

shown in Figure 10 (red) for time periods 𝜏 = 1 second (1 second is the initial wave period, 1/𝑓 ! ). For a robotic worm in a straight pipe controlled to a traveling wave shape, the order of magnitude of the net normal forces is relatively constant over time. Thus, it would be possible to predict the average total !

normal force !!  !! ! !! !!!! 𝑁!! ∆𝑡 ! /𝜏 from the static deformation. Note that the forces in (18) could come from any type of ground contact, so even though our simulation is in a pipe, it may be possible to get similar results with ground walking. When on the ground and not in a pipe, the sum of the normal forces is approximately equal to the weight and (16) is equivalent to the frictional cost of transport in (Alexander 2003, equation 6.6, page 91) times two because of the top and bottom contacts. Further, we find that a better approximation of the energy losses is to use only the normal forces of the segments corresponding to the non-maximal actuator forces, which, in the case of the SHC controller, are the segments where the SHC node activation is not close to 1: !!

Non-­‐anchoring  Friction  COT  Estimate   =

2𝜇!"#$#%& 𝑀  

!!

𝑁!! ∆𝑡 ! /𝜏                                  (19) !  !! ! !! !  !{!!! ,  !!! }  

  Where 𝐴!! and A!! are the two anchoring segments, defined as the segments with the most compressive actuator force, −𝐹!! , at any given time. In other words, we can make a more efficient robot by ensuring that the segments that are not anchoring are not dragging along the pipe. The actual value of the COT from (17) can be greater than the estimate in (19) if there is slip at the anchoring segments. It can be greater than both estimates (18) and (19) if there is some slip backwards and the same distance must be crossed repeatedly. Alternatively, the COT (17) can be less than (19) if the segments progress most when putting the least pressure on the ground. The frictional COT for (C) as calculated by (17) and estimated by both (18) and (19) is plotted in Figure 10. Finally, we compare all three controllers, (A), (B), and (C), in Figure 11. This demonstrates that adding feedback to both the force control and the timing control of a worm-like robot can improve performance. The total energy lost to slip against friction of Controller C is 60% of that of Controller A over the course of the 1.6 m long pipe. The COT varies along the pipe. Calculating the slope of the data in Figure 11, the highest COT for controller C was 30% of the highest COT for Controller A and the steadystate COT of Controller C was 25% of the steady-state of Controller A.

Figure 8. Key state variables for steady locomotion in a straight pipe under controller (C). Contact forces (normal force shown here) and segment shape (percent elongation plotted here) are determined from actuator forces. For controller (C), the actuator forces are generated by two modes, anchor and progression. The anchor mode (equations 11-13) acts when a segment’s associated SHC node (in gray) is active. For the segments associated with inactive nodes, the progression mode sets the actuator force to a weighted integral of the SHC nodes (equation 14). In turn, the contact forces trigger the growth of SHC node activity, keeping node activation and contact forces in phase. Note that when new contacts occur there can be oscillations in normal force between segments. In this plot, peak contact forces are limited to fit between the lines for each segment where necessary. All parameter ranges are listed in Table 3.

 

Figure 9. The simulated worm robot with Controller (C) shown when the third node from the posterior (segment 3 in Figure 4) is in the middle of its active period. The contact forces are dark bars, colors as in Figure 4. Vertical lines indicate the contact points of this segment (their spacing is the stride length). Note that the stride length starts out at 16 cm, and becomes smaller as the contact length increases. Simulation times and brief descriptions are provided at right.

 

Figure 10. Figure 10. Simulation energy results for a simulated worm robot going through a pipe narrowing using controller (C). (Left) The system energy, averaged over the wave period, starts at zero and increases to a stable value. (Middle) The actuator input work approximates the loss due to slip plus net system energy. This serves as a validation of our simulation and indicates that mitigating slip could greatly improve energy efficiency. (Right) Numerical costs of transport (COT) (work per center of mass progression per mass) calculated over wave periods (equations 17-19). The worst case would be the sum of the normal forces times the friction coefficient (equation 18). However, if the anchoring segments do not slip, and the other segments bear little ground loading, we can obtain a lower COT estimate. In this case, the friction coefficient times the sum of the mean normal forces borne by all but the segments with largest negative actuator force is a good predictor for the cost of transport as calculated from the slip work (equation 19). Also shown is the inertial COT from (Alexander 2003, p. 91, eq. 6.7), which, as expected, is small (ranging from .8 to 1.5 m/sec/sec).

 

Figure 11. Comparison of three different controllers in a pipe narrowing. (Top) The force, 𝐹!! , applied by the actuator of a single segment (segment 6, see Figure 4), for each of the three controllers. Controller (A) forces are unaffected by the narrowing. Controller (B) forces have modified amplitudes and Controller (C) forces have modified amplitudes and periods. This latter adaptation gives (C) better performance as is shown in the next two panels. (Center) The mapping between time and center-of-mass (COM) progress. The reciprocal of the slope is the speed. Controller (C) has the most consistent and fastest speed through the narrowing. (Bottom) Progress vs. slip loss. The slope over the mass is the COT. Each controller can be efficiently tuned for a straight fixed-diameter pipe. However, when the pipe changes diameter, controller (C) performs the best. The location of the pipe narrowing is labeled on the x-axis, but the narrowing effect extends beyond the two lines because the anterior segments enter the narrowing before the COM and the posterior segments leave the narrowing after the COM.

 

7. Conclusions and Future Work Coupling radial and longitudinal deformation is effective for the locomotion of soft-bodied earthworms. Like our simulation, radial pressures of earthworms vary with segment radius (Quillin 1998) and peristaltic waves involve multiple progressing segments between anchoring segments (Gray & Lissmann 1938). Our work suggests that peristaltic locomotion may be more efficient than previously believed because the limiting cost of transport is more closely related to the ground pressure on the moving segments (which can be minimized, even for isotropic friction) than the total ground pressure (which must exceed the weight and be sufficient to anchor the body). Minimizing slip makes robotic worm applications in search and rescue, medicine, and inspection even more promising. This type of locomotion has modeling challenges that we have overcome for one set of parameters. We show that this radial-extension coupling, combined with a goal to reduce slip, constrains the placement of the contact points between waves. What is controllable are the contact forces, the shape of the non-contacting segments, and the relative timing at each contact point. For efficient locomotion, we control the anchoring segments to generate normal forces and control the progressing segments to maintain the inter-contact length while progressing. Understanding these two distinct control modes, and when to transition between them, is better than seeking a good open-loop controller, especially if the environment is not known a priori, as shown in our example of a changing diameter pipe. Other examples of changing environments might be changes in friction coefficient, such as in a wet or dusty area, or changes in loading, such as when a tunnel changes incline. Adaptive control might work well in compliant tubes since substrate compliance reduces efficiency and reducing friction forces wherever possible may mitigate tube deformation losses in stride length (Zarrouk, et al. 2010; Zarrouk & Shoham 2012b). For simulations in more complex environments (e.g., 3-D, uneven or loose terrain, and compliant tubes), more detailed approximations of the body and the underlying actuators driving the soft-bodied robot may be required (Jones & Walker 2006; Renda, et al. 2012). Implementing this control on a robot requires individual segment actuation and sensory feedback, such as from normal force sensors (Liu, et al. 2006) or artificial skin sensors (Park, et al. 2012) or mechanism measurements (Park, et al. 2011). There should be a minimum of two segments (one to extend and one to retract) between any pair of anchoring segments (Figure 4). To explore narrowings, the segments should be able to decrease as much as possible in height and also adapt the number of segments involved in each wave. Since contact points are constrained posterior to anchoring segments, decisionmaking should start at the front and propagate backwards. From this simulation, we can see that segmental coordination in peristalsis is critical. Uncoordinated motion wastes energy, will not result in forward progress, and can get the body in a state that is increasingly difficult to maneuver out of (e.g., getting stuck with contact points lodged too close together in a narrowing). Sufficient coordination for a specific task can be accomplished without feedback as in Softworm or our sine wave controller. However, with feedback, the simulated worm robot can do the same task more efficiently (in our case with 40% less energy lost to slip, and 25% of maximum the frictional cost of transport) and has the potential to accomplish a wider variety of goals. The neuroethology of earthworms can provide valuable control inspiration. The stochastic dynamic oscillator we chose, stable heteroclinic channels, can be considered an abstract neural population model with a pattern of neural connectivity related to the matrix in (8). Our work is an example where adding an SHC improves performance. But there is more to learn from biology. For example, the experiments by (Moore 1923; Gray & Lissmann 1938; Collier 1939; Quillin 1998) and others may be used to further refine and inspire control of speed and dynamic adjustment of wave number and spacing, as well as explore new behaviors such as burrowing, as demonstrated by (Omori, et al. 2011), turning, and object manipulation. 8. Acknowledgements This work was supported by NSF research Grant No. IIS-1065489.

 

9. Appendix: Simulation Mechanics 9.1. Development of Equations of Motion First, we show how the equations of motion are set up using the principle of virtual work with d’Alembert’s principle to relate the effects of forces applied in different directions. The energy terms for the work against body springs and inertia are: !

 

!!

! !!  ! !

!!!

𝑆(𝑙) 𝑑𝑙! +   !!  !

1 𝑘𝜃 ! + 2 !

!

!!  !

1 𝑚 𝑥! !                                                                                          (A1) 2

where 𝑆 𝑙 describes a nonlinear spring attached horizontally between the nodes with rest length 𝑙 ! and current length 𝑙, 𝑘 describes a torsion spring at angle 𝜃 between rigid rhombus links in contacting rhombuses, 𝑚 is the lumped mass at a node, the length of the rhombus links is 𝑙 ! , and 𝑛 is the number of segments (the number of degrees of freedom is 𝑛 + 1). See Figure 2. Further, the work done by the ground contact forces and control forces is defined by: !

!!  !

d𝑥! + 𝑑𝑥!!! 2𝑇! − 2

!

!

𝑁! dℎ! − !!  !

𝐹! dℎ!                                                                                      (A2) !!  !

where 𝑇! and 𝑁! are the horizontal and vertical ground reaction forces at each element and 𝐹! is the force from the controller compressing the rhombuses. (Note that where the pipe is not a constant-diameter pipe the ground reaction forces can be rotated to horizontal and vertical directions based on the surface pitch). Using d’Alembert’s principle (Lanczos, 1970; Meirovitch, 1970) we take the variation of the internal energy (A1) and set that equal to the variation of work done by external forces (A2). !

!

𝑆 𝑙! 𝛿𝑙! +  𝑘𝜃! 𝛿𝜃! + 𝑚𝑥! 𝛿𝑥! = !!  !

!!!

1 𝑇! 𝛿𝑥! + 𝑇! 𝛿𝑥!!! − 𝑁! 𝛿ℎ! − 𝐹! 𝛿ℎ!              (A3) 2

Next we need relationships between the different variables and their variations (𝛿𝑙, 𝛿𝑥, 𝛿ℎ, 𝛿𝜃). We can see from Figure 2 that 𝑙! =   𝑥! −   𝑥!!! , taking the variation gives 𝛿𝑙! =  𝛿𝑥! −   𝛿𝑥!!!                                                                                                                                                            (A4) The heights of the rhombuses (the maximum diameters of the worm segments) are related to the ! ! 𝑙 ℎ lengths via the Pythagorean theorem: ! 2 + ! 2 =   𝑙 ! ! . Thus, a change in virtual height, 𝛿ℎ,  can be related to a virtual change in length, 𝛿𝑙, as 2𝑙! 𝑙! 𝛿ℎ! =   − 𝛿𝑙! =   − 𝛿𝑥! −   𝛿𝑥!!!                                                                                            (A5) 2ℎ! ℎ! Finally, we can relate 𝜃 to 𝑥 using definition of cosine 𝑙!!! 𝑙! 𝜃! = cos !! − cos !!                                                                                                              (A6) ! 2𝑙 2𝑙 ! Taking the variation and substituting from (A5), then (A4) 1 1 𝛿𝜃! = − 𝛿𝑙!!! + 𝛿𝑙!                                                                                                   4 𝑙!

!

!

!

−   𝑙!!!   4 𝑙 ! ! −   𝑙!   1 1  = − 𝛿𝑙!!! + 𝛿𝑙!                                                                                                                                                                           ℎ!!! ℎ! 𝛿𝑥!!! 𝛿𝑥! 𝛿𝑥! 𝛿𝑥!!! =   − + + −                                                                                                              (A7) ℎ!!! ℎ!!! ℎ! ℎ!

  Using (A4), (A5), and (A7) we can now write (A3) in terms of only variations in 𝛿𝑥 s (𝛿𝑥! , 𝛿𝑥! , 𝛿𝑥! , 𝛿𝑥! … 𝛿𝑥! ): !

𝑆 𝑙!

𝛿𝑥!!! −   𝛿𝑥! + 𝑘𝜃! −

!!  !

𝛿𝑥!!! 𝛿𝑥! 𝛿𝑥! 𝛿𝑥!!! + + −   + 𝑚𝑥! 𝛿𝑥!                                             ℎ!!! ℎ!!! ℎ! ℎ!

!

   =

𝑇! 𝛿𝑥! +   𝛿𝑥!!!   + 𝑁! + 𝐹! !!!

𝑙! 𝛿𝑥! −   𝛿𝑥!!!                                          (A8) ℎ!

(A8) is one equation, the sum of many terms multiplied by each node’s virtual displacement 𝛿𝑥! . Each virtual displacement 𝛿𝑥! is an independent degree of freedom, so their values are arbitrary. Therefore all the coefficients of 𝛿𝑥! must sum to zero in order for (A8) to be true for any virtual displacement  𝛿𝑥! . Thus we want to group the coefficients of each virtual displacement ( 𝛿𝑥! , 𝛿𝑥! , 𝛿𝑥! , 𝛿𝑥! … 𝛿𝑥! ) to obtain 𝑛 + 1 different equations indexed by 𝑗 ∈ 0, … , 𝑛 , as in D’Alembert’s form of the principle of virtual work (Lanczos, 1970). In other words, the indices are changed so that there are no 𝛿𝑥!!! or 𝛿𝑥!!! terms, only 𝛿𝑥! , which can be factored out, leaving us equation (3), duplicated here: 𝜃!!! 𝜃! 𝜃! 𝜃!!! −𝑆 𝑙! + 𝑆 𝑙!!! + 𝑘 − + + −   + 𝑚𝑥!   ℎ! ℎ!!! ℎ! ℎ!!! 𝑙! 𝑙!!! = 𝑇! +   𝑇!!! + 𝑁! + 𝐹! − 𝑁!!! + 𝐹!!!                            (3) ℎ! ℎ!!! where such terms are in the range of the index (for example, there is no 𝑙! or 𝑙!!! , so 𝑆 𝑙! = 0 ). The motion of each node depends on that of the neighboring nodes. If 𝑘 is nonzero nodes ±2 away affect the acceleration of a node. 𝑆 is the spring force as a function of length given by (2). Now we have specified a differential equation for each degree of freedom where F is known (the actuator input), T and N are zero where the simulated worm robot body is not in contact with the pipe, but unknown if in contact (ground contact and friction constraints are explained below and in Table 2). Due to these extra unknowns, this is not a system of ordinary differential equations. However, this is a solvable system of nonlinear equations if we use the following approximation for a dynamic term, at each time 𝑡 shown as a superscript here: 𝑥! ! − 𝑥! !!! 𝑥! !!! − 𝑥! !!! ! !!! −   𝑥! − 𝑥! 𝑥! ! 𝑥! !!! 𝑥! !!! 𝑥! !!! ∆𝑡 ! ∆𝑡 !!! 𝑥! ! ≅ ≅   =   − − +    (A9) ∆𝑡 ! ∆𝑡 ! ∆𝑡 ! ∆𝑡 ! ∆𝑡 ! ∆𝑡 ! ∆𝑡 ! ∆𝑡 !!! ∆𝑡 ! ∆𝑡 !!! Note that if the ground is not orthogonal to the simulated worm robot, as in the case of a tapering pipe, a simple rotation of the ground angle, tan(d𝑟/d𝑥), can be performed on 𝑇 and 𝑁. 9.2. Equation Solver To solve, we need to minimize the net absolute error, 𝑒! , where 𝑒! is defined as the error in (3), the difference between the 𝑚𝑥! term and all the other terms. The lengths of elements on the ground are known functions of the pipe diameter. Friction constraints are applied for sliding contact. All the unknowns can be considered in one vector, 𝑢. 𝑙, which  are  off  the  ground 𝑁, which  are  appled  on  the  ground 𝑢 =                                                (A10) 𝑇, which  are  applied  when  not  slipping COM, if  all  points  are  free For all the unknowns (the lengths that are not fixed, 𝑤, the normal forces if the contact state is not equal to zero, 𝑁 (0 = free in Table 2), the tangential forces if contact state is equal to one, 𝑇! , and COM if

  no segment is in contact) we numerically determine how a small change (d𝑙, d𝑁, d𝑇, dCOM) of random sign affects error vector. Note that if there is slip, a change in 𝑁! results in a recalculation of 𝑇! . When put together, and divided by the small changes, the result is a numerical approximation of the gradient of the error

!!!

!!!

. If the system was linear, we could solve for ∆𝑢 in a single pass with Matlab’s matrix algebra

functions such as left matrix divide. 𝜕𝑒! 𝜕𝑒! 𝜕𝑒! … 𝜕𝑢! 𝜕𝑢! 𝜕𝑢!!! 𝑒! ∆𝑢! 0 𝜕𝑒! 𝜕𝑒! 𝜕𝑒! 𝑒! ∆𝑢! … =   0                          (𝐴11) 𝜕𝑢! 𝜕𝑢!!! ⋮ + 𝜕𝑢! ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 𝑒!!! ∆𝑢!!! 0 𝜕𝑒!!! 𝜕𝑒!!! 𝜕𝑒!!! … 𝜕𝑢! 𝜕𝑢! 𝜕𝑢!!! Since our system is nonlinear, this calculation step is run iteratively until a solution is found. Note that bounds on the unknowns must be imposed so that lengths are positive and less than 2𝐿. The error is calculated after adjusting 𝑢 = 𝑢 + ∆𝑢 from (A11), then if the error is less, Δ𝑢 is reduced by 10%. If the error is not improved by the solution from (A11), gradient descent can be used, where the gradient terms are weighted by current error values (so as to prioritize equations with worst errors first). 𝜕𝑒! 𝜕𝑒! 𝜕𝑒! ! … 𝜕𝑢! 𝜕𝑢! 𝜕𝑢!!! 𝑒! ∆𝑢! 𝜕𝑒! 𝜕𝑒! 𝜕𝑒! 𝑒 ∆𝑢! ! … = −   𝜕𝑢! 𝜕𝑢! 𝜕𝑢!!! ⋮                              (𝐴12) ⋮ ⋱ ⋮ ⋮ ⋮ 𝑒!!! ∆𝑢!!! 𝜕𝑒!!! 𝜕𝑒!!! 𝜕𝑒!!! … 𝜕𝑢! 𝜕𝑢! 𝜕𝑢!!! If there is still no improvement in the error value 𝜕𝑢 is reduced by 1%, and another round of calculations are performed. 9.3. Coulomb Friction Constraints The solutions must be found that satisfy one of the sets of force constraints in Table 2. We assume that contact state is the same as the previous state and solve with the associated constraint in Table 2. If the check in column three of Table 2 is satisfied, the next timestep is taken. If the checks are not satisfied, a new contact state vector is tried. We find that properly ordering these checks of contact state is important for solving quickly. First, any previously free segments that penetrate the pipe are considered. Those segments that were previously close to the pipe and now penetrate the pipe the furthest have their states changed to either: state 3 if the penetrating contact position moved backward from previous timestep, state 4 if the penetrating contact position moved forward from previous timestep, state 1 if the penetrating contact position is unchanged within (1e-4)l0, or state 2 if the penetrating contact position is unchanged within (1e-4)l0 and previous timestep state = 0 and 1 has been tried but no set of constraints was found to satisfy the equations. This last condition is the physical equivalent of an orthogonal impact. If proportional friction forces are calculated, they are very sensitive to small inaccuracies in contact location. State 3 is rare (Figure 12), but the inclusion of this state prevents the simulation from stalling when the equations get stiff. Any attached segments that are generating unphysical contact forces (such as tangential forces in the same direction as slipping or negative normal forces) are reset to the 0 free state for recalculation. If the tangential forces for a state 1 sticking contact are too large, the contact state is set to 3 or 4 based on the direction of the

  forces. This state updating process is iterated several times. If the constraints are not satisfied, the results are unused and the timestep is halved and the process starts again from the previous constraint-satisfying timestep values. In Figure 12, each line represents the worm’s segments at a single timestep. Contact states are colorcoded with (Table 2). The timesteps are large when only two segments are in contact. However, when adjacent segments are contacting, timesteps must be reduced.

 

Figure 12. The primary output of the simulation is a time series of mass node locations (black wavy lines) and ground friction forces. Each segment consists of two adjacent nodes and is color-coded by its ground friction state (see legend). When a segment is not contacting the ground at all (the free contact state), it is shade is scaled by the segment length. (Left) The clipped sine wave controller (A) is advancing the worm from left to right in a constant-diameter pipe. A regular pattern is established by the end of the pipe. The nodes rarely move backwards and each ground contact is an almost vertical column (in gray) because there is little slip. When contacts on adjacent segments coincide (vertical columns overlapping in time) one or both segments will slip (blue or pink). At these times, the simulation time-step also decreases. (Middle) There is a narrowing in the pipe, 0.7 ≤ x ≤ 1. Either, controller (A) (shown) or controller (B) causes a regular pattern through the pipe, but the contacts slip more (the vertical columns are now leaning stacks due to sliding friction). The columns are also wider because the contact length is longer when the pipe radius is narrower. The segments are in contact for greater time periods (and thus the columns of gray are taller). (Right) The SHC controller with physical feedback (C) reduces slip in a pipe with variable diameter by breaking the regular traveling wave pattern. The columns of gray, pink, and blue contact are shorter and straighter. While the pattern looks less regular, it is in fact more efficient, as can be observed from the COM progress (orange line is straighter).

  References Afraimovich V, Tristan I, Huerta R and Rabinovich M I 2008 Winnerless competition principle and prediction of the transient dynamics in a Lotka-Volterra model Chaos 18 043103 Afraimovich V S, Zhigulin V P, and Rabinovich, M I 2004 On the origin of reproducible sequential activity in neural circuits Chaos 14 1123–1129 Alart P and Curnier A 1991 A mixed formulation for frictional contact problems prone to Newton like solution methods Computer Methods in Applied Mechanics and Engineering 92 353–375 Alexander RM 2003 Alternative Land Locomotion in Principles of Animal Locomotion 86–102 Allotey N and El Naggar MH 2008 Generalized dynamic Winkler model for nonlinear soil–structure interaction analysis Canadian Geotechnical Journal 45 560–573 Asari V, Kumar S and Kassim, IM 2000 A fully autonomous microrobotic endoscopy system Journal of Intelligent and Robotic Systems 28 325–341 Ashwin P and Karabacak O 2011 Robust Heteroclinic Behaviour, Synchronization, and Ratcheting of Coupled Oscillators Dynamics, Games and Science II 2 125–140 Azad M and Featherstone R 2010 Modeling the contact between a rolling sphere and a compliant ground plane Proc Astralasian Conf. Robotics and Automation Bender J and Schmitt A 2006 Constraint-based collision and contact handling using impulses Proc Int Conf on Computer Animation and Social Agents Bertetto A M and Ruggiu M 2001 In-pipe inch-worn pneumatic flexible robot IEEE/ASME Int Conf on Advanced Intelligent Mechatronics 1226–1231 Boxerbaum A S, Chiel H J and Quinn R D 2009 Softworm: A Soft, Biologically Inspired Worm-Like Robot Neuroscience Abstracts, Society for Neuroscience Annual Meeting, Chicago, IL USA Boxerbaum A S, Chiel H J and Quinn R D 2010 A new theory and methods for creating peristaltic motion in a robotic platform Proc IEEE Int Conf on Robotics and Automation 1221–1227 Boxerbaum A S, Horchler A D, Shaw K M, Chiel H J and Quinn R D 2011 A controller for continuous wave peristaltic locomotion Proc IEEE Int Conf on Robotics and Automation 197–202 Boxerbaum A S, Daltorio K A, Chiel H J and Quinn, R D 2012 A Soft-Body Controller With Ubiquitous Sensor Feedback Proc 1st Int Conf on Living Machines Boxerbaum A S, Shaw K M, Chiel H J and Quinn R D 2012 Continuous wave peristaltic motion in a robot International Journal of Robotics Research 31 302–318 Boxerbaum A S 2012 Continuous Wave Persistalic Motion in a Robot, PhD Dissertation Case Western Reserve University Boyle J H, Berri S and Cohen N 2012 Gait Modulation in C elegans: An Integrated Neuromechanical Model Frontiers in Computational Neuroscience 6 10 Chatterjee A 1997 Rigid Body Collisions PhD Dissertation Cornell University Cheng N, Ishigami G, Hawthorne S, Hansen M, Telleria M, Playter R and Iagnemma K 2010 Design and analysis of a soft mobile robot composed of multiple thermally activated joints driven by a single actuator Proc IEEE Int Conf on Robotics and Automation 5207–5212 Collier H O J 1939 Central Nervous Activity in the earthworm Journal of Experimental Biology 16 286– 299 Conradt J and Varshavskaya P 2003 Distributed central pattern generator control for a serpentine robot Proc Int Conf Artificial Neural Networks 2–5 Cowan L S and Walker I D 2008 “Soft” Continuum Robots: the Interaction of Continuous and Discrete Elements Proc 11th Int Conf on the Simulation and Synthesis of Living Systems 126–133 Crespi A and Badertscher A 2005 An Amphibious robot capable of snake and lamprey-like locomotion Robotics and Autonomous Systems 50 Dario P, Carrozza M C and Pietrabissa A 1999 Development and in vitro testing of a miniature robotic system for computer-assisted colonoscopy Computer Aided Surgery 4 1–14 Degallier S, Righetti L, Gay S and Ijspeert A J 2011 Toward simple control for complex, autonomous robotic applications: combining discrete and rhythmic motor primitives Autonomous Robots 31 155–181

  Dorgan K M 2010 Environmental Constraints on the Mechanics of Crawling and Burrowing Using Hydrostatic Skeletons Experimental Mechanics 50 1373–1381 Fukunaga A S, Morookian J M, Quillin K, Stoics A and Thakoor S 1998 Earthwormlike Exploratory Robots JPL New Technology Report NPS-20266 Gause G F 1932 Experimental Studies on the Struggle for Existence J Exp Biol 9 389–402 Gilpin M E 1975 Limit Cycles in Competition Communities The American Naturalist 109 51–60 Glass P, Cheung E and Sitti M 2008 A legged anchoring mechanism for capsule endoscopes using micropatterned adhesives IEEE Trans on Bio-medical Engineering 55 2759–67 Gmiterko A, Kelemen M, Virgala I, Surovec R and Vacková M 2011 Modeling of a Snake-like Robot Rectilinear Motion and Requirements for its Actuators Proc 15th IEEE Int Conf on Intelligent Engineering Systems 91–94 Gray B Y J and Lissmann H W 1938 Studies in Animal Locomotion VII: Locomotory Reflexes in the Earthworm in Journal of Experimental Biology 15 518–521 Guckenheimer J and Holmes P 1988 Structurally stable heteroclinic cycles Math Proc Camb Phil Soc 103 189–192 Hirose S and Mori M 2004 Biologically Inspired Snake-like Robots Proc IEEE Int Conf on Robotics and Biomimetics 1–7 Hoeg H D, Slatkin A B, Burdick J and Grundfest W 2000 Biomechanical modeling of the small intestine as required for the design and operation of a robotic endoscope Proc IEEE Int Conf Robotics and A 1599– 1606 Horchler A D 2013 SHCTools: Matlab ToolBox for Simulation, Analysis, and Design of Stable Heteroclinic Channel Networks, Version 1.1 GitHub http://github.com/horchler/SHCTools, accessed Apr. 2013 Ijspeert A J 2008 Central pattern generators for locomotion control in animals and robots: a review Neural Networks 21 642–53 Ijspeert A J, Crespi A, Ryczko D and Cabelguen J-M 2007 From swimming to walking with a salamander robot driven by a spinal cord model Science 315 1416–20 Ingram M E 2006 Whole Skin Locomotion Inspired by Amoeboid Motility Mechanisms: Mechanics of the Concentric Solid Tube Model Masters Thesis, Virginia Polytechnic and State University Jones B A and Walker I D 2006 Kinematics for multisection continuum robots IEEE Trans on Robotics 22 43–55 Kassim I, Ng W S, Feng G, Phee S J, Dario P, Martin P and Mosse C A 2003 Review of Locomotion Techniques for Robotic Colonoscopy Proc IEEE Int Conf on Robotics and Automation 1086–1091 Keller J B and Falkovitz M S 1983 Crawling of worms Journal of Theoretical Biology 104 417–442 Keudel M and Schrader S 1999 Axial and radial pressure exerted by earthworms of different ecological groups Biology and Fertility of Soils 29 262–269 Kiebel S J, Von Kriegstein K, Daunizeau J and Friston K J 2009 Recognizing sequences of sequences PLoS Computational Biology 5 e1000464 Lanczos C 1970 The Variational Principles of Mechanics 4th Ed. General Publishing Co, Canada Laurent G, Stopfer M, Friedrich W, Rabinovich M I, Volkovskii A and Abarbanel H D I 2001 Odor Encoding as an Active, Dynamical Process: Experiments, Computations and Theory Ann Rev Neuro 24 263–297 Li D, Cross M C, Zhou C and Zheng Z 2012 Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles Phys. Rev. E 85 016215 Lin H T and Trimmer B 2010 The substrate as a skeleton: ground reaction forces from a soft-bodied legged animal Journal of Experimental Biology 213 1133–42 Lissmann H W 1950 Rectilinear Locomotion in a Snake (boa occidentalis) Journal of Experimental Biology 26 368–379 Liu W, Menciassi A, Scapellato S, Dario P and Chen Y 2006 A biomimetic sensor for a crawling minirobot Robotics and Autonomous Systems 54 513–528

  Lu Y 2013 Survey of Multibody Dynamics Simulation Packages Renselear Polytechnic Institute CS Robotics Lab http://cs.rpi.edu/~trink/sim_packages.html, accessed Apr. 2013 Mangan E V, Kingsley D A, Quinn R D and Chiel H J 2002 Development of a peristaltic endoscope Proceedings 2002 IEEE International Conference on Robotics and Automation 347–352 May R and Leonard W 1975 Nonlinear aspects of competition between three species SIAM Journal of Applied Mathematics 29 243–253 Mazzolai B, Margheri L, Cianchetti M, Dario P and Laschi C 2012 Soft robotic arm inspired by the octopus. II. From artificial requirements to innovative technological solutions Bioinspir & Biomim 7 025005-1–14 McInnes C and Brown B 2010 A dynamical systems approach to micro-spacecraft autonomy Proc 20th AAS/AIAA Space Flight Mechanics Meeting 136 1199–1218 Meirovitch L 1970 Methods of Analytical Dynamics, McGraw-Hill Menciassi A, Gorini S, Pernorio G and Dario P 2004 A SMA Actuated Artificial Earthworm Proc IEEE Int Conf on Robotics and Automation 3282–3287 Moore A R 1923 Muscle tension and reflexes in earthworm The Journal of General Physiology 5 327–333 Murakami Y, Uchiyama H, Kurata J and Maeda M 2006 Dynamical locomotion analysis and a model for the peristalic motion of earthworms SICE-ICASE Int Joint Conf 4224–4229 Nakazato Y, Sonobe Y and Toyama S 2010 Development of an in-pipe micro mobile robot using peristalsis motion Journal of Mechanical Science and Technology 24 51–54 Neves F S and Timme M 2012 Computation by Switching in Complex Networks of States Phys Rev Lett 109 018701 Neves F S and Timme M 2009 Controlled perturbation-induced switching in pulse-coupled oscillator networks J Phys A: Math Theor 42 345103 Niebur E and Erdos P 1991 Dynamics of undulatory progression on a surface Biophys Journal 60 1132– 1146 Nowotny T and Rabinovich M I 2007 Dynamical Origin of Independent Spiking and Bursting Activity in Neural Microcircuits Phys Rev Lett 98 128106 Omori H, Nakamura T and Yada T 2009 An underground explorer robot based on peristaltic crawling of earthworms Industrial Robot: An International Journal 36 358–364 Omori H, Murakami T, Nagai H, Nakamura T and Kubota 2011 Planetary Subsurface Explorer Robot with Propulsion Units for Peristaltic Crawling IEEE Int Conf on Robotics and Automation 649–654 Onal C D, Chen X, Whitesides G M and Rus D 2011 Soft mobile robots with on-board chemical pressure generation Int Symposium on Robotics Research Park J, Hyun D, Cho W-H, Kim T-H and Yang H-S 2011 Normal-Force Control for an In-Pipe Robot According to the Inclination of Pipelines IEEE Transactions on Industrial Electronics 58 5304–5310 Park Y-L, Chen B-R and Wood R J 2012 Design and Fabrication of Soft Artificial Skin Using Embedded Microchannels and Liquid Conductors IEEE Sensors 12 2711–2718 Quillin K 1998 Ontogenetic scaling of hydrostatic skeletons: geometric, static stress and dynamic stress scaling of the earthworm lumbricus terrestris The Journal of experimental biology 201 1871–1883 Rabinovich M, Volkovskii A, Lecanda P, Huerta R, Abarbanel H D I and Laurent G 2001 Dynamical encoding by networks of competing neuron groups: Winnerless competition Phys Rev Lett, 87 068102 Rabinovich, M, Huerta R, Varona P and Afraimovich V S 2008 Transient cognitive dynamics, metastability, and decision making PLoS Computational Biology 4 e1000072 Renda F, Cianchetti M, Giorelli M, Arienti A and Laschi C 2012 A 3D Steady State Model of a TendonDriven Continuum Soft Manipulator Inspired by Octopus Arm Bioinspir & Biomim 7 025006-1–12 Saga N and Nakamura T 2004 Development of a peristaltic crawling robot using magnetic fluid on the basis of the locomotion mechanism of the earthworm Smart Materials and Structures 13 566–569 Schulke M, Hartmann L and Behn C 2011 Worm-like locomotion systems: development of drives and selective anisotropic friction structures Proc 56th Int Scientific Colloquim Schwebke S and Behn C 2013 Worm-like robotic systems: Generation, analysis and shift of gaits using adaptive control Artificial Intelligence Research 2 12–35

  Seok S, Onal C D, Wood R, Rus D and Kim S 2010 Peristaltic Locomotion with Antagonistic Actuators in Soft Robotics Proc IEEE Int Conf on Robotics and Automation 1228–1233 Shaw K M, Lu H, McManus J M, Cullins M J, Chiel H J and Thomas P J 2010 Evidence for a central pattern generator built on a heteroclinic channel instead of a limit cycle Front. Neurosci. Conference Abstract: Computational and Systems Neuroscience 2010 Shaw K M, Park Y-M, Chiel H J and Thomas P J 2012 Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit SIAM Journal on Applied Dynamical Systems 11 350–391 Shaw K M, Cullins M J, Lu H, McManus J M, Thomas P J and Chiel H J 2012 Investigating localized sensitivity in the feeding patterns of Aplysia californica Front. Behav. Neurosci. Conference Abstract: 10th Int. Congress of Neuroethology Slatkin A B, Burdick J and Grundfest W 1995 The development of a robotic endoscope Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems 2 162–171 Spardy L E, Markin S N, Shevtsova N A, Prilutsky B I, Rybak I A and Rubin J E 2011 A dynamical systems analysis of afferent control in a neuromechanical model of locomotion: II. Phase asymmetry J. Neural Eng 8 065004 Steigenberger 2003 Contribution to the mechanics of worm-like motion systems and artificial muscles Biomechanics and Modeling in Mechanobiology 37–57 Steigenberger and Abeszer 2008 Quasistatic inflation processes within rigid tubes ZAMM (Journal of Applied Mathematics and Mechanics) 556–572 Stone E and Holmes P 1990 Random Perturbations of Heteroclinic Attractors SIAM Journal of Applied Mathematics 50 726–743 Tesch M, Lipkin K, Brown I, Hatton R, Peck A, Rembisz J and Choset H 2009 Parameterized and Scripted Gaits for Modular Snake Robots Advanced Robotics 23 1131–1158 Transeth A A, Leine R I, Glocker C, Pettersen K Y and Liljebäck P 2008 Snake Robot Obstacle-Aided Locomotion: IEEE Trans on Robotics 24 88–104 Transeth A A, Pettersen K Y and Liljebäck P 2009 A survey on snake robot modeling and locomotion Robotica 27 999 Trimmer B A, Takesian A E, Sweet B M, Rogers C B, Hake D C and Rogers D J 2006 Caterpillar locomotion: A new model for soft-bodied climbing and burrowing robots Proc Int Symposium on Technology and the Mine Problem Trivedi D, Rahn C D, Kier W M and Walker I D 2008 Soft robotics: Biological inspiration, state of the art, and future research Applied Bionics and Biomechanics 5 99–117 Tucker V A 1970 Energetic cost of locomotion in animals Comparative Biochemistry and Physiology 34 841–6 Vaidyanathan R, Chiel H J and Quinn R D 2000 A hydrostatic robot for marine applications Robotics and Autonomous Systems 30 103–113 Varona P, Rabinovich M I, Selverston A I and Arshavsky Y I 2002 Winnerless competition between sensory neurons generates chaos: A possible mechanism for molluscan hunting behavior Chaos 12 572– 677 Verduzco F and Alvarez J 2000 Homoclinic Chaos in 2-DOF Robot Manipulators Driven by PD Controllers Nonlinear Dynamics 21 157–171 Wadden T, Hellgren J, Lansner A and Grillner S 1997 Biological Cybernetics Intersegmental coordination in the lamprey: simulations using a network model without segmental boundaries Biological Cybernetics 76 1–9 Wang K and Yan G 2007 Micro robot prototype for colonoscopy and in vitro experiments Journal of Medical Engineering and Technology 31 24–8 Wang W, Wang Y, Qi J, Zhang H and Zhang J 2008 The CPG control algorithm for a climbing worm robot Proc 3rd IEEE Conf on Industrial Electronics and Applications 675–679

  Webster V A, Lonsberry A J, Horchler A D, Shaw K M, Chiel H J and Quinn R D 2013 A Segmental Mobile Robot with Active Tensegrity Bending and Noise-driven Oscillators Proc IEEE/ASME Int Conf on Advanced Intelligent Mechatronics (submitted) Wilson H R and Cowan J D 1972 Excitatory and inhibitory interactions in localized populations of model neurons Biophysical Journal 12 1–24 Wriggers P 1995 Finite Element Algorithms for Contact Problems Arhives of Computational Methods in Engineering 2 1–49 Wu X and Ma S 2010 CPG-based control of serpentine locomotion of a snake-like robot Mechatronics 20 326–334 Yamamoto H, Sekine Y, Sato Y, Higashizawa T, Miyata T, Iino S, Ido K and Sugano K 2001 Total enteroscopy with a nonsurgical steerable double-balloon method Gastroinest Endosc 53 216–220 Yamashita A, Matsui K, Kawaniski R, Kaneko T, Murakami T, Omori H, Nakamura T and Asama H 2011 Self-Localization and 3-D Model Construction of Pipe by Earthworm Robot Equipped with OmniDirectional Rangefinder Proc IEEE Int Conf on Robotics and Biomimetics 1017–1023 Zarrouk D, Sharf I and Shoham M 2010 Analysis of Earthworm-like Robotic Locomotion on Compliant Surfaces Proc IEEE Int Conf on Robotics and Automation 1574–1579 Zarrouk D, Sharf I and Shoham M 2012 Energy Analysis of Worm Locomotion on Flexible Surface Proc IEEE Int Conf on Intelligent Robots and Systems Zarrouk D and Shoham M 2012a Analysis and Design of One Degree of Freedom Worm Robots for Locomotion on Rigid and Compliant Terrain Journal of Mechanical Design 134 Zarrouk D and Shoham M 2012b Conditions for Worm-Robot Locomotion in a Flexible Environment: Theory and Experiments. IEEE Transactions on Bio-medical Engineering 59 1057–67 Zimmermann K, Zeidis I and Behn C 2009 Mechanics of Terrestrial Locomotion Springer Fachmedien Wiesbaden, Germany Zuo J, Yan G and Gao Z 2005 A micro creeping robot for colonoscopy based on the earthworm Journal of Medical Engineering and Technology 29 1–7